self-assembly in nafion membranes upon hydration: water...

Self-Assembly in Nafion Membranes upon Hydration: Water Mobilityand Adsorption IsothermsAleksey Vishnyakov and Alexander V. Neimark*

Department of Chemical Engineering, Rutgers, the State University of New Jersey, 98 Brett Road, Piscataway, New Jersey 08854,United States

*S Supporting Information

ABSTRACT: By means of dissipative particle dynamics (DPD) and Monte Carlo(MC) simulations, we explored geometrical, transport, and sorption properties ofhydrated Nafion-type polyelectrolyte membranes. Composed of a perfluorinatedbackbone with sulfonate side chains, Nafion self-assembles upon hydration andsegregates into interpenetrating hydrophilic and hydrophobic subphases. Thissegregated morphology determines the transport properties of Nafion membranesthat are widely used as compartment separators in fuel cells and other electrochemicaldevices, as well as permselective diffusion barriers in protective fabrics. We introduced acoarse-grained model of Nafion, which accounts explicitly for polymer rigidity andelectrostatic interactions between anionic side chains and hydrated metal cations. In aseries of DPD simulations with increasing content of water, a classical percolationtransition from a system of isolated water clusters to a 3D network of hydrophilicchannels was observed. The hydrophilic subphase connectivity and water diffusion werestudied by constructing digitized replicas of self-assembled morphologies and performing random walk simulations. A non-monotonic dependence of the tracer diffusivity on the water content was found. This unexpected behavior was explained by theformation of large and mostly isolated water domains detected at high water content and high equivalent polymer weight. UsingMC simulations, we calculated the chemical potential of water in the hydrated polymer and constructed the water sorptionisotherms, which extended to the oversaturated conditions. We determined that the maximum diffusivity and the onset offormation of large water domains corresponded to the saturation conditions at 100% humidity. The oversaturated membranemorphologies generated in the canonical ensemble DPD simulations correspond to the metastable and unstable states of Nafionmembrane that are not realized in the experiments.

I. INTRODUCTION

Polyelectrolyte membranes (PEMs) are widely used ascompartment separators in electrochemical devices.1 Inparticular, Nafion (DuPont, Figure 1), a perfluorinated linearpolymer with hydrophilic sulfonate side chains, is the basicmaterial for proton conducting membranes of fuel cells.1 Whilethe acid form of Nafion is most common for electrochemicalapplications, metal-substituted membranes are also of interestas permselective diffusion barriers in protective fabrics.2 Uponhydration, PEMs are known to undergo nanophase segregationinto hydrophilic and hydrophobic subphases.1 In the case ofNafion, the former contains hydrophilic side chains, counter-ions, and sorbed water; the latter is comprised of theperfluoroalkane backbone. At higher hydration levels, thehydrophilic subphase is continuous and provides facile watertransport.The transport properties of hydrated PEM depend on their

nanostructure, which is determined by polymer chemistry andsolvent content. In Nafion and other similar polyelectrolytes(e.g., Flemion, Aquivion), the segregation morphology isirregular with no particular long-range order of hydrophilicaggregates.3 The irregular structure complicates both inter-pretation of SAXS and SANS results and theoretical/simulation

predictions of structural and transport properties. It should benoted that the Nafion equivalent polymer weight Meq, sidechain length, and distribution of side chains along the skeletonare not precisely controlled during polymer synthesis. Thisuncertainty further complicates the comparison of modelingefforts with available experimental data, even on a qualitativelevel.Macroscopic models typically present the Nafion perfluor-

oalkane backbone as a continuum medium where spherical orcylindrical water aggregates, which grow in size and coalesce asthe water activity increases (e.g., refs 4−6, review 7). Theelectrostatic interactions are modeled by double layers formedat the cluster surfaces by side chains on the outer side and bycounterions on the inner side. Such models can reasonablydescribe water sorption and ion exchange in Nafion,6,8 but theyinvolve adjustable parameters fitted directly to the experimentalresults and presume a particular geometry of hydrophilicaggregates in advance. The self-consistent field theory allowed amore detailed consideration of polyelectrolyte morphology, but

Received: May 20, 2014Revised: August 26, 2014Published: August 26, 2014

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it has been limited to linear block copolymers9,10 presented asGaussian chains. The same applies to the mesoscale dynamicdensity functional theory of polyelectrolytes,11,12 which wasalso used to model Nafion segregation.13 A more complicatedapproach was implemented by Galperin and Khokhlov,14 whoeffectively divided the branched polymer into “subchains”whose conformations may be considered independent of eachother. The subchain conformations were treated statisticallyusing probability distributions, and the system of interactingsubchains was approximated by a single ideal chain in a self-consistent field. This approach was implemented on a lattice,producing a sponge-like irregular network of hydrophilicaggregates.14

Atomistic molecular dynamics (MD) simulations of Nafionstarted with minimization of individual side chains.15 In ourearlier works,16,17 we studied solvation of Nafion oligomers inwater and other solvents. These simulations confirmedmicrophase segregation on the scale of a few nm and formationof hydrophilic clusters linked by “merging” bridges, which buildup and snap off due to thermal fluctuations. However, the sizeof the simulated system was insufficient to analyze the clusternetwork morphology. As the atomistic and united-atom forcefields were established and computational facilities grew, MDsimulations became able to handle systems substantially largerthan the size of individual hydrophilic aggregates and started totackle the problems of the membrane segregation morphologyand water transport in perfluorinated and other ionomers.18−33

Generally, atomistic simulations produced a qualitativelyevident picture of individual water clusters that coalesce andform a continuous water cluster network as hydration increases.Even with modern computers, computational expenses severelylimit the spatial and temporal scales of PEM simulations. Forthis reason, Knox and Voth34 used in their MD simulations ofNafion the initial configurations corresponding to differentmorphologies suggested in the experimental literature. Thelarge size of the system allowed for performing virtual SAXSscattering experiments with the simulated polymer config-urations. The authors concluded that the “ionomer peak”present in scattering results is insensitive to the segregationmorphology, that further complicates the interpretation ofSAXS and SANS results. Extensive MD simulations wereemployed to look into more subtle issues, such as the influenceof equivalent weight, side chain length, and distribution alongthe skeleton on structural and transport properties of hydratedNafion,34−37 gas adsorption,38 Nafion behavior near solidsurfaces,39 transport under electrostatic field,35 and influence ofadded nonvolatile solvents such as phosphoric acid and ionicliquids.40,41 In the most recent work, Daly et al.42 performed anup to 200 ns long atomistic NPT ensemble MD simulation ofwater self-diffusion in Nafion and calculated water adsorptionisotherms with GPU based MC simulations. This work sets abenchmark for current atomistic simulations; however, a

limited size of simulated systems (up to 9 nm) is stillinsufficient to investigate the membrane segregated morphol-ogy.A significant increase in spatial and temporal scales was

achieved with coarse-graining and reducing the system degreesof freedom by “lumping together” several atoms to formquasiparticles or “beads”.43 In the most popular for modelingpolymeric systems coarse-grained MD (CGMD) method,quasiparticles interact through effective pairwise hard-corepotential (such as Lennard-Jones) parameters that arecommonly derived from atomistic modeling.13,44−47 In orderto draw reliable conclusions about the morphological structureof self-assembled polymer, the size of the simulation box mustexceed the characteristic scale of segregation by at least 1 orderof magnitude. Malek et al.48 modeled a Nafion−carbonnanocomposite using Lennard-Jones potentials betweenquasiparticles representing different fragments of Nafionpolymer, solvent, and carbon particles. The authors obtaineda self-assembled structure composed of the hydrophobicbackbone and water clusters qualitatively similar to thatfound in earlier atomistic MD simulations. However, the largersystem size allowed the evolution of segregated morphology tobe distinguished as hydration progressed. At low water content,hydrophilic domains were roughly spherical and poorlyconnected. At higher hydration, a sponge-like network ofroughly cylindrical aggregates of 3 nm in diameter was formed.In this work, we employ the dissipative particle dynamics

(DPD) method,49,50 which implies soft short-range repulsionpotentials and therefore allows for much longer time steps(compared to MD) and facilitates the system equilibration.Several authors employed DPD simulations for modelingNafion-type membranes. The first DPD simulation of Nafion inacid form was conducted by Yamamoto et al.51 Theconservative repulsion parameters were estimated from themixing energy calculations conducted with atomistic modeling.The electrostatic interactions were implicitly mimicked byshort-range forces.51−53 The authors found irregular segrega-tion morphologies, with reasonable correspondence toexperimental results. Later, Dorenbos et al.54,62 and Wu etal.53,55 employed the same model for studies of nanostructureand water diffusion in several perfluorinated ionomers thatdiffered by Meq and side chain length. Sawada et al.

56 accountedfor possible cross-linking of the perfluorinated skeleton chainsand found that this effect leads to much smaller hydrophilicaggregates of only 1.8 nm in diameter. Eliott et al.57 combinedDPD results with experimental SAXS/SANS studies using anovel model-independent procedure. The modeling revealed amultilevel membrane organization, with hydrophilic−hydro-phobic segregation on a smaller scale and larger scaleorganization of the fluorocarbon backbone. This result isconsistent with previous NMR studies.58,59 Jorn and Voth60

modeled the segregation in Nafion with DPD and calculated

Figure 1. Dissection of Nafion monomer into beads (example with Meq = 944 Da shown). Beads are connected by nearest neighbor (1−2) andsecond neighbor (1−3) bonds to control polymer rigidity and side chain flexibility.

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proton conductivity using the smoothed particle hydro-dynamics approach, based on local concentration and chargedensities in the resulting structures. The calculated conductiv-ities showed very reasonable agreement with experimental data.The DPD studies mentioned above made an important step

forward in modeling the segregation morphology of Nafion.However, these models lacked several features that are criticallyimportant for analyses of structural and transport properties ofPEM: polymer rigidity and explicit electrostatic interactions.Noteworthy, parametrization of these models was based ongeneric DPD parameters devised for aqueous systems.49,50,61 Inthis paper, we suggest a novel DPD model of hydrated metal-substituted Nafion membranes. The proposed model accountsfor polymer chain rigidity that is different for the Nafionbackbone and side chains and includes explicit electrostaticforces between the charged polymer fragments and dissociatedcounterions. In contrast to previous works, we attempted tocustomize the coarse-grained interaction parameters against theavailable experimental and atomic simulation data. For the firsttime, we not only generated self-assembled structures inhydrated Nafion membranes and explored the specifics ofnanosegregated morphologies, but we also established theirthermodynamic properties, in particular, the relationshipbetween the level of hydration and humidity.This work builds upon our earlier DPD model developed for

studies of interactions of toxic chemicals with perm-selectiveNafion barriers.62 The paper is structured as follows. In sectionII, we formulate the DPD model of coarse-grained Nafionpolymer and justify its dissection into the soft repulsive beads.Section III discusses the DPD model parametrization that isbased on a combination of atomistic MD and coarse-grainedMC simulations. The evolution of self-assembled morphologyas the water content increases is studied in section IV. Thehydrophilic network connectivity and water diffusion areexplored in section V by using digitized replicas of the systemsnapshots and modeling random walk motion of a tracer. Insection VI, using the MC test particle insertion method, wedetermine the chemical potential of adsorbed water at givenhydration levels and construct the water sorption isotherms,which allowed us to identify the conditions of saturation and todetermine the limits of stability of generated conformations.Main conclusions and suggestions for prospective research aregiven in section VII.

II. COARSE-GRAINED MODEL OFMETAL-SUBSTITUTED NAFION

The proposed coarse-grained model of Nafion is constructedfollowing the classical DPD approach originated from theseminal work of Groot and Warren.49,50,61 The system underconsideration is presented as a multicomponent mixture ofbeads mimicking characteristic fragments on the hydratedpolymer, that interact via pairwise conservative soft repulsive,harmonic bond, and electrostatic forces, as well as random andvelocity dependent drag forces:

= + + +

+

F r F r F r F r F r

F r v

( ) ( ) ( ) ( ) ( )

( , )

ij ij ij ij ij ij ij ij ij ij

ij ij ij

(C) (B) (E) (R)

(D)(1)

All beads are assigned an equal effective diameter Rc. The softrepulsion force Fij

(C) acts between overlapping beads: Fij(C)(rij) =

aIJ(1 − rij/Rc)rij/rij at r < Rc, Fij(C)(rij) = 0 at r ≥ Rc, where aIJ is

the repulsion parameter specific to the given bead pair of types

I and J. Following the standard approach to DPD simulations ofself-assembly,50 the intracomponent repulsion parameters aIIbetween beads of the same type are set equal, irrespective to thebead type. The beads are tightly packed with a substantialoverlap. We accepted the reduced bead packing density of ρRc

3

= 3, common in DPD simulations.50

The random and drag force also acted between overlappingbeads along the line connecting the bead centers. Random forceFij(R) that accounts for thermal fluctuations is taken proportional

to the conservative force that is also acting along the vectorbetween the bead centers: Fij

(R)(rij) = σwRrijθij(t)rij, where θij(t)is a randomly fluctuating in time variable with Gaussianstatistics. The drag force is velocity-dependent: Fij

(D)(rij, vij) =−γwD(rij)(rij*vij), where vij = vj − vi and vi and vj are the currentvelocities of the particles. We assume the common relationshipbetween the drag and random force parameters wD(r) =[wR(r)]2 = (1 − r/Rc)

2 at r < Rc, wD(r) = 0 at r ≥ Rc. σ and γ are

parameters that determine the level of energy fluctuation anddissipation; they are related as σ2 = 2γkT, which allows aconstant temperature to be maintained in the course ofsimulation via the Langevin thermostat. We assumed γ = 4.5, acommon value fitted to the diffusion coefficient of water.The polymer beads are connected by harmonic bonds

Fij(B)(rij) = Kb(rij − r0)rij/rij, where Kb is the bond rigidity, which

depends on the bead type, and r0 is the equilibrium length.Following our recent papers,63,64 in addition to this nearestneighbor (1−2) bond, we also introduced the second neighbor(1−3) harmonic bonds in order to control the skeleton rigidityand side chain flexibility.

Bead Types. We consider metal-substituted Nafionoligomers of chemical structure shown in Figure 1. Theequivalent weight Meq of the polymer was varied from 944 (12carbon atoms between the neighboring side chains) to 1744(28 carbon atoms between the neighboring side chains); Meqvalues are given for the anion (the total Meq can be obtained byadding the mass of the cation). In all simulations, the sidechains are separated by equal fluorocarbon fragments, and theside chains were of the same length. In order to explicitlyinclude electrostatic interactions in the DPD model, wedissected the Nafion chain using four bead types. Thehydrophobic beads of type C, which lump together fourcarbon atoms with attached fluorine atoms, represent skeletonperfluoroalkane fragments (CFx)4. The side chain perfluor-oether fragment (O−CF(CF3)−CF2−O−) is modeled by thebead of type E that is less hydrophobic than bead C. Thenecessity of such detailization is caused by the role that theperfluoroether fragment plays in interactions of Nafion withphosphororganic chemicals.58,65 Two bead types S− and Srepresent negatively charged dissociated CF2CF2SO3

− andneutral non-dissociated CF2CF2SO3M sulfonate groups,respectively. The latter includes an alkali metal and has to bedistinguished at low hydration levels where water content isinsufficient for dissociation of all sulfonate groups. Watermolecules were lumped into hydrophilic beads of type W.Hydrated alkali metal cations were represented by chargedbeads of type M+, which in addition to the cation includedseveral water molecules.

Bead Volumes. The adopted dissection of the polymer intocoarse-grained beads is performed to minimize the difference involumes between the polymer fragments represented bydifferent bead types. The effective volumes of fragments areestimated from the functional group volumes (“Bonditables”66,67 that are available for perfluorinated compounds68



but not for sulfonates), as well from the molecular volumes ofrepresentative compounds calculated with the PQS ab initiopackage.69 To provide for an approximate equality with thevolumes of the skeleton and sulfonate beads, the volume ofwater bead W is chosen as 135 Å3, which corresponds to theeffective bead size of Rc = 7.4 Å. With a bead density of ρRc

3 =3, the corresponding number of water molecules in the W beadis nW = 4.5. Note that this noninteger number of molecules perquasiparticle is permissible in coarse-grained DPD modeling.The bead types, respective fragments, and their volumes arelisted in Table 1. The underlying calculations of effectivevolumes are described in the Supporting Information, section I.Electrostatic Interactions. Electrostatic interactions be-

tween the dissociated sulfonate beads S− and hydratedcounterion M+ beads are treated using the smeared-chargeapproach of Groot.70 Instead of point charges interacting viathe Coulomb potential that diverges at r → 0, each charge wasdistributed within the sphere of the smearing diameter Re =1.2Rc with the charge density decreasing linearly from the beadcenter toward the periphery.70 The choice of Re is ratherarbitrary; since the charge cloud size does not have a clearphysical meaning,71 its influence on the thermodynamicproperties of electrolytes is still to be examined. We chose Resmaller than recommended by Groot in an attempt to improvecalculation efficiency.71 The comparison between the screenedand Coulomb potential created by two point charges is given inthe Supporting Information. The sulfonate S− and counterionM+ beads bear charges of −e and +e, respectively. Partialcharges on the skeleton and perfluoroether fragments areneglected. Ewald summation was employed to account for thelong-range electrostatic contribution.The hydrated counterion bead M+ is chosen to contain 1

cation and 3.5 water molecules and had the same parameters asthe water beadW, except for being positively charged. Thus, weeffectively assumed that the volume of a counterion is close tothat of one water molecule, which is reasonable for largercounterions. Merging of water molecules with an alkali metalion in the same bead has important implications. In our model,the bead charges are fixed; that is, S beads were not allowed todissociate into cation M+ and anion S− beads in the course ofsimulation and, respectively, M+ and anion S− beads cannotrecombine. Water content in Nafion is expressed either as watermass per unit mass of dry polymer or by the hydration level λexpressed as the number of water molecules per sulfonate

group. Since eachM+ bead included 3.5 molecules of water, λ =3.5 is the minimum hydration level at which all counterionsmay be considered as dissociated from their sulfonate groupsand modeled as M+ beads. At lower hydration levels, λ < 3.5, allwater molecules are assigned to M+ beads; i.e., there are nouncharged W beads. To secure electroneutrality, the number ofcharged sulfonate group S− beads is equal to the number ofcounterions M+, and the number of neutral S beads iscalculated from the difference. At λ > 3.5, all sulfonate groupsare assumed to be dissociated.

III. PARAMETERIZATION OF INTERACTIONPOTENTIALS

Intracomponent Repulsion Parameter. We assumed thesame intracomponent (“self-repulsion”) parameter aII = 50kT/Rc for all interactions between beads of the same type. This self-repulsion parameter is chosen to approximately reflect theoverall compressibility of hydrated Nafion that is much higherthan the pure water compressibility (e.g., compare 1.8 × 10−9

Pa−1 for n-perfluorononane72 with 4 × 10−10 Pa−1 for water). Itis worth noting that the chosen value of aII is lower than thevalue of 119.8kT/Rc that would be estimated from the purewater compressibility (e.g., ref 50, Supporting Information toref 73), as was implied in previous DPD models of Nafion.51

This is a significant difference: solvated membrane in ourmodel is much “softer”.

Perfluoroalkane Skeleton Rigidity. The Nafion skeletonis modeled as a linear sequence of hydrophobic beads C, eachrepresenting four CFx groups. Skeleton rigidity affects thedistance between the neighboring side chains and the overallelasticity of the matrix. In order to fit bond length and rigiditiesthat are determined by the torsional potentials rather then bythe stiffness of covalent bonds, we performed MD simulationsof a perfluorohexadecane melt with force field from ref 74similar to our prior simulations of Nafion31,32 (see theSupporting Information, section II, for simulation details).Simulations were conducted in the NPT ensemble atatmospheric pressure and a temperature of 450 °C, sinceC16F34 freezes under ambient conditions. The MD trajectorieswere recorded to disk. Each molecule was dissected into fourfragments (four CFx groups each), and probability distributionsof intramolecular distances between the centers of mass of eachfragments were calculated (Figure 2). After that, we conductedDPD simulations of tetramers of skeleton C beads under

Table 1. Coarse-Grained Model of Hydrated Nafion: Bead Types, Interaction Parameters, Bond Lengths, and Stiffness

parameterization of coarse-grained model of Nafion

conservative parameter, aIJRc/kT

bead type fragment effective volume (Å3) W K+ S S− C E

1 W (H2O)4.5 134.5 50 50 50 50 65 602 M+ [M(H2O)3.5]

+ 122.1a 50 50 50 50 65 605 S −(CF2)2SO3M 136.5a 50 50 50 50 75 656 S− −(CF2)2SO3

− 119.3 50 50 50 50 75 653 C −(CFx)4− 131.3 65 65 75 75 50 594 E −OCF2−CFCF3O− 117.1 60 60 65 65 59 50

nearest neighbor bonds U1−2(r) = 0.5K1−2(r − re)2 second neighbor bonds U1−3(r) = 0.5K1−3(r − re)

2

bond K1−2Rc2/kT re (Å) bond K1−3Rc

2/kT re (Å)

C−C 160 4.1 C−C 80 8.2C−E 160 3.7 C−E 30 6.0E−S 160 4.4 C−S 30 6.0

aEstimated for K+ counterion.



similar conditions and fitted the nearest neighbor and secondneighbor bond parameter paying most attention to distancesbetween beads separated by two and three bonds. The resultingdistributions are shown in Figure 2. The agreement betweenDPD and MD 1−3 and 1−4 distance distributions is veryreasonable. We decided not to attempt achieving a goodagreement between MD and DPD distributions for the distancebetween the neighboring beads in the chain and instead limitthe nearest neighbor skeleton bond stiffness to K1−2 = 160kT/Rc

2. Fitting the 1−2 distance distribution requires very stiffnearest neighbor bonds,75 which leads to much shorter timesteps and drastically slows down the simulations. At the sametime, the nearest neighbor bond stiffness hardly affects theeffective distance between side chains and therefore is ofsecondary importance for our purpose.Perfluoroalkane−Water Repulsion Mismatch Parame-

ter. ΔaCW was obtained by performing DPD simulation of n-C36F74 that was presented as a 9-mer of C beads with thenearest neighbor and second neighbor bond parameters derivedfrom MD simulation of perfluorohexadecane. The mismatchparameter ΔaCW = 15kT/Rc was determined by fitting to arespective atomistic MD simulation. The results are shown inFigure 3. At low values of ΔaCW, coarse-grained n-C36F74behaves as an extended chain. At ΔaCW= 15kT/Rc, the DPDsimulation shows frequent transitions between the collapsedglobule and extended chain configurations, and at ΔaCW >20kT/Rc, the globular configurations prevail. Since the MDsimulation was not long enough to estimate the time the chainspends in the coil and globule configurations, the extended andcollapsed states were modeled separately. The end-to-enddistances of the extended states obtained by MD and DPD arein good agreement, thus justifying the choice of both ΔaCW andbond parameters.Side Chain Repulsion Parameters. The mismatch

parameters for perfluoroether fragment E are estimated fromthe water and skeleton parameter approximately taking intoaccount that two fluorocarbon groups are replaced by

hydrophilic oxygens. As a result, they are modeled as mildlyhydrophobic ΔaEW = 9kT/Rc. The sulfonate end-groups areconsidered as hydrophilic (ΔaSW = 0) beads, whetherdissociated or not. The sulfonate groups were assumed tointeract very unfavorably with the other fragments of thepolymer, and mild mismatch of ΔaEW = 9kT/Rc was assigned tothe repulsion between C and E beads representing the skeletonand the ether side chain fragment, respectively.

Side Chain Bonds. Using the short-range conservativerepulsion parameters described above, we obtained the nearestneighbor and second neighbor bond parameters for the sidechain from MD and DPD simulations of a single Nafionmonomer (depicted in Figure 1) in a water bath. Figure 4shows the distribution of distances between DPD beads and

Figure 2. Fitting the bond rigidity in the DPD model ofperfluorohexadecane C16F34 to the results of atomistic MD simulationsat T = 450 K. The distributions of distances between the chain DPDbeads separated by one, two, and three harmonic bonds are matchedto the distributions of distances between the centers of mass of thecorresponding fragments of the atomistic chain, each of which containsfour CF3 or CF2 groups. Reasonable agreement is obtained for beadsseparated by two and three bonds.

Figure 3. Results of MD and DPD simulations of C36F44 in water at T= 300 K. The distributions of distances between DPD beads separatedby three, five, and eight harmonic bonds are compared with thedistributions of distances between the centers of mass of thecorresponding fragments in the atomistic representation, each ofwhich contains four CF3 or CF2 groups.

Figure 4. Results of MD and DPD simulations of C36F44 in water at T= 300 K. The distributions of distances between S, E, and C beads arefitted to the distributions of distances between the centers of mass ofthe corresponding fragments in the atomistic representation, shown inFigure 2.






respective distributions of the intramolecular distances betweenthe centers of mass of corresponding Nafion fragmentsobtained in MD simulations. The presence of a side chainsubstantially changes the conformations of the skeleton.Although distorted trans C−C−C−C torsion angles continueto dominate the skeleton conformation similarly to perfluor-oalkane chains (Figure 3), we also observed hairpin-likeconformations, where the skeleton makes a sharp bend closeto the point of the side chain attachment. As a result, thedistribution of the distances between the fragments correspond-ing to second neighbor fluorocarbon beads has a minor secondpeak, which we could not reproduce with crude DPD models.Nevertheless, we found the agreement between MD and DPDresults reasonable, taking into account the coarse-grainednature of the model under consideration. The bond parametersare presented in Table 1.

IV. DPD SIMULATIONS OF SELF-ASSEMBLEDMORPHOLOGY OF HYDRATED NAFION

Using the DPD model described above, we studied the self-assembled phase segregation in metal-substituted Nafionmembranes by equilibrating the system in the canonicalensemble DPD simulation. We considered the polymer ofdifferent equivalent weight Meq and varied the water content toexplore the evolution of membrane morphology uponhydration. The equivalent weight Meq of the polymer isdetermined by the number of beads between the neighboringside chains. Simulations were performed in a cubic 30 × 30 ×30Rc

3 cell containing 81 000 beads, starting from a randomconfiguration. The method of Paganabarra et al.76 was used tointegrate the equations of motion with a time step of 0.02. Thebalance of energy fluctuation and dissipation provided aLangevin thermostate that allowed maintaining the averagetemperature within 1% of the required value and temperaturefluctuations below 5%. The segregation progress wascharacterized by monitoring the number of pairs of overlappingW beads. When this number stabilized, equilibrium wasconsidered established. Further details of DPD simulationsmay be found in the Supporting Information, section III.

We considered Nafion polymer with side chains separated by12, 16, 20, and 28 skeleton beads, which corresponded toMeq =944, 1144, 1344, and 1744 Da for the anion, respectively. It isworth noting that most published data that dealt with alkalimetal substituted Nafion membranes are of Meq ≈ 1100−1200Da. Experimental water sorption isotherms on these mem-branes were summarized in ref 65. At 100% humidity, watersorption decreases from approximately 29% wt in Li+ form to4−6% wt in Cs+ form. The K+ form of Nafion, which wastargeted in our study, is in the middle of this range; it adsorbs9.5−12% of dry polymer weight, which corresponds to λ =6.2−7.9, according to refs 65, 77, and 78. Water sorptiondecreases with the Nafion equivalent weight Meq due to theincreasing fraction of hydrophobic skeleton groups.4 In theNPT DPD simulations, we considered a wide range ofhydration levels from λ = 2.25 to λ = 13.5.Figure 5 shows the snapshots of selected systems of different

Meq and λ. In general, the evolution of the system morphologyis qualitatively similar to the classical scenario of the percolationsystem formation.4,51,79 At low hydration levels, the water andcounterions form small clusters around the sulfonate groups(note that in our model at λ < 3.5 there is not enough water todissociate all sulfonate groups). As hydration increases, apercolation transition occurs: isolated water clusters grow andcoalesce, forming an irregular 3D network of worm-likechannels. Further increase in hydration leads to formation ofinterconnected spheroidal clusters that grow in size. Thesystem morphology beyond the percolation threshold resem-bles the classical model proposed by Gierke and Hsu,4 whoassumed a regular network of spherical water clusters up to 4nm in diameter connected by cylindrical channels ofapproximately 1 nm in diameter. It is worth noting thatnarrow (2−3 water molecules in diameter) cylindrical waterbridges between water clusters were observed in our atomisticsimulations of Nafion.65 As the water content increases further,spherical clusters grow and form large spheroidal aggregates,signifying macroscopic separation of the system into water andhydrated Nafion phases. Below, we show that thesemorphologies correspond to supersaturated states that would

Figure 5. Snapshots of nanophase separation in hydrated Nafion: Meq = 1144 (top) and 1744 (bottom). Hydration level λ = 2.25 (a, e), 6 (b, f),9 (c, g), and 13.5 (d, h). Nafion skeleton and perfluoroether side chain beads are shown in red, sulfonate groups in dark blue, counterions in green,and water beads in light blue.




not be observed under real experimental conditions withhumidity limited by saturation.

V. CONNECTIVITY OF THE HYDROPHILIC SUBPHASEAND WATER DIFFUSIVITY

The visual, purely qualitative observations of Nafion morphol-ogy do not allow evaluation of aqueous subphase connectivity,which determines the transport of water through PEM and isextremely important for most applications of these materials. Itshould be noted that the common DPD implementationemployed in this work is unable to simultaneously reproduceexperimental self-diffusion and viscosity in liquid systems.50

Our DPD model of Nafion is designed to predict the structureof segregated polyelectrolyte, but it is not capable of directlypredicting the absolute values of diffusion coefficients. Toquantify the effect of hydrophilic subphase connectivity onwater diffusion, we constructed digitized replicas of thesimulated configurations in a fashion similar to that suggestedby Dorenbos et al.54 For each equilibrated simulation, wecollected a trajectory of 400 frames. The simulated config-uration was mapped onto a cubic lattice grid. The step of thegrid in each dimension was 0.5Rc, which is about the size of onewater molecule. Since DPD beads overlap, one lattice site maybelong to several beads of different type. In order to assign agiven lattice site to either a mobile hydrophilic or immobilehydrophobic subphase, we calculated the site preference asfollows:

∑ =

= − | − | | − |

< = | − | ≥

=p r t w r r

w r r r r R r r

R w r r R

( ) ( , ),

where ( , ) (1 / ) at

, 0 at

li

N

i i l

i l i l i l

i l

1

c2

c c

Here, rl is the radius-vector to the center of lattice site l, ri is theradius-vector to the ith bead, and ti is a mobility coefficientrelated to the bead type: ti = −1 for all mobile beads (that is,water and hydrated counterions) and ti = 1 was assigned to allpolymer beads (C, E, and S bead types). Note that the M beadmostly represents water and is a part of the water path throughthe system, while S/S− beads are not, despite its hydrophilicnature. The entire S bead represents a fragment of polymer andincludes hydrophobic CF2 groups. Water may hydrate the Sbead but does not diffuse through. p(rl) shows whether mobileor immobile beads prevail in the close vicinity of site l. If p isnegative, site l is assigned to a mobile (hydrophilic) subphase;otherwise, site l is assigned to the immobile subphase. Threecharacteristic examples of such obtained digitized latticereplicas are shown in Figure 6.Having created digitized lattice replicas of the membrane

morphology, we modeled water diffusion within the hydrophilicsubphase as a simple random walk of a tracer particle within thereplica’s mobile phase. We therefore assumed that the structuralevolution of the segregated polymer is much slower than thewater diffusion inside the hydrophilic subphase. Each randomwalk started from a randomly selected lattice site that belongedto the hydrophilic subphase, and each step was an attemptedmove to one of the six sites that neighbored the currentlocation. The move was accepted if the attempted site belongedto the mobile subphase. 104 random walks were performed oneach of 400 replicas for proper averaging. We calculated themean square displacement (MSD) between the current

location and the original location as a function of the numberof steps.Characteristic examples of MSD dependences are shown in

Figure 7. Line 1 is characteristic of a system with isolated water

clusters below the percolation threshold: the tracer never leavesthe cluster in which the random walk starts. Above thepercolation threshold, the MSD (lines 2 and 3) asymptoticallyconverges onto a straight line, which is characteristic for adiffusion-type process along a continuous network of waterclusters. The slope of this line characterizes the water diffusivityin the segregated membrane of given hydration λ relative to thewater diffusivity in the bulk water modeled as a random walk ona simple cubic lattice. As such, the ratio of this slope to that forthe random walk on an unrestricted lattice (all sites belong tothe hydrophilic subphase) quantifies the ratio DW/DP of thewater self-diffusion coefficients in Nafion and in bulk water.The relative water diffusion coefficients, DW/DP, calculated in

random walk simulations are presented in Figure 8 and TableS3 of the Supporting Information. Water diffusivity in themembrane is significantly reduced compared to bulk water, and

Figure 6. Digitized lattice replicas of the membrane morphologymapped onto a cubic lattice (hydrophilic subphase shown in lightpink): (a) Meq = 1144, λ = 9; (b) Meq = 1744, λ = 13.5.

Figure 7. Mean square distance as a function of the number of stepsfor the tracer random walk within the hydrophilic subphase ofhydrated Nafion for several characteristic configurations. (1) Meq =1144, λ = 2.25: water forms small isolated clusters, diffusion islocalized. (2) Meq = 1144, λ = 9: this level of hydration is about theexperimental value at 100% humidity and corresponds to fastestdiffusion obtained for given Meq. Water forms a well-definedcontinuous subphase. (3) Meq = 1344, λ = 13.5: the mean squaredisplacement reflects large water aggregates and their poorconnectivity in unstable configurations; fast diffusion at short timeand slow diffusion at longer time scale are observed.





DW progressively decreases with the increase of Meq. DW issmaller than DP by 1 order of magnitude for Meq = 944 and by2 orders of magnitude for Meq = 1144 and is negligibly small forMeq = 1344, and larger. Interestingly, such a big reduction ofthe diffusion coefficient compares very well with the valuesderived by Rivin and Schneider (see ref 78) from themembrane permeation measurements80,81 for K+ substitutedNafion 112 in contact with liquid water; see Figure 8.At λ = 2.25, water formed a percolating network in some of

the recorded configurations only in Nafion of the lowestmolecular weight. In other systems, the aqueous subphaseconsisted of small and isolated clusters or individual counterionbeads. It should be noted that the random walks wereperformed on static configurations; thus, the effects of dynamicpercolation that involves slow migration of water clusters andformation of temporary bridges between water clusters found inatomistic MD simulations78 are not considered with thistechnique.A continuous mobile subphase manifested by nonzero

diffusion coefficients was formed above the percolationthreshold at λ = 4.5 in all systems but Meq = 1744. With

further increase of hydration, the water mobility increases,achieving a maximum. For Nafion of Meq = 1144, the maximumis achieved around λ = 9 and corresponds to a water diffusivity2 orders of magnitude smaller than that in the bulk. It isinteresting that such a big reduction of the diffusion coefficientcompares very well with the values derived by Rivin andSchneider (see ref 78) from the membrane permeationmeasurements80,81 for K+ substituted Nafion 112 in contactwith liquid water; see Figure 8.A non-monotonic dependence of the diffusion coefficient on

the saturation, which was pronounced in all systems weconsidered, seems to be counterintuitive, yet it is explained bythe specifics of the hydrophilic subphase morphology at highhydration. Indeed, the growth of large disconnected waterclusters described in section IV (Figure 6), in which the tracergets trapped, leads to the decrease of the hydrophilic subphaseconnectivity. It is visible on the snapshots (Figure 6b) for highMeq: large spherical clusters are not connected to each other.To some extent, this “in silico” observation may be an artifact ofthe static polymer matrix: in a system of large clusters, thepossibility of the formation of temporal intercluster bridgesshown in MD simulations16 may lead to additional pathways forwater diffusion that cannot be captured with static replicas. It isworth noting that, as shown below, the membraneconformations obtained in NPT simulations at high hydrationlevels, which exhibit big and poorly connected water clustersand, respectively, declining diffusion coefficients, correspond tooversaturated thermodynamic conditions and, thus, are notdealt with in practice.Data in Figure 8 shows a dramatic dependence of the water

diffusion on the molecular weight Meq. The water clusternetwork connectivity naturally declines with Meq; since thefraction of hydrophilic beads decreases, the distances betweenthe side chains increase, thus making the water clusters moreisolated. This effect is very pronounced: DW decreases morethan an order of magnitude as Meq increases from 944 to 1344and by another order of magnitude at Meq = 1344. For Meq =1744, we obtained zero DW at all hydration levels above λ =6.75. A weaker dependence of DW onMeq was obtained in DPDsimulations of Dorenbos et al.54 We are not aware of a detailedexperimental analysis of Dw−Meq dependencies for nonacidforms of Nafion, but metal-substituted Flemion membranes

Figure 8. Self-diffusion coefficient of water in hydrated Nafion reducedto the diffusion coefficient of pure water for polymers of hydrationlevel λ for equivalent molecular weightMeq of 1144 and 1344, obtainedwith the random walk simulation. The green filled diamonds areexperimental data for K+ substituted Nafion-112 membranes (Meq ≈1100 Da).78 Similar dependences for other Meq are shown inlogarithmic scale in the Supporting Information. The error bars wereestimated from the diffusion coefficients for each of the individualNafion snapshots.

Figure 9. (a) Water sorption isotherms calculated using Widom MC trial insertion into the configurations recorded from DPD simulations of Nafionof different equivalent polymer weights. Solid lines serve as guides to the eye. The configurations at a > 1 are thermodynamically unstable andcorrespond to oversaturated conditions; phase separation into liquid water and hydrated Nafion does not happen in simulations due to the limitedsystem size and periodic boundary conditions. Lines are drawn as guides to the eye. The black square shows the experimental value for K+

substituted Nafion withMeq ≈ 1200 Da.77,80 (b) Water activity and reduced diffusion coefficient as functions of the water content forMeq = 1144 Da.The maximum of water mobility is achieved near saturation and declines in the region of oversaturation due to the formation of disconnected waterdomains.





show an even weaker (compared to ref 54) decline of diffusivitywith the equivalent weight.82

VI. CALCULATION OF WATER SORPTION ISOTHERMSIN HYDRATED NAFION

The large water aggregates obtained in DPD at higherhydration levels raise a general question regarding theinterpretation of the results of mesoscale modeling in thepolymers that may absorb a limited amount of solvent.Although DPD simulations performed in the canonicalensemble reveal the morphology of hydrated polymer, theygive no direct information on the equilibrium water activity towhich a particular hydration level corresponds. Furthermore,we do not even know whether the polymer at a given hydrationequilibrated as a closed system in the canonical ensemble(especially at high hydration levels which are of interest forprotective membranes) is thermodynamically stable undergiven environmental conditions, i.e., as an open system in thegrand canonical ensemble.To estimate the water isotherm and to study the system

stability, we calculated the chemical potential of water in theDPD generated hydrated membranes. For a given hydrationlevel λ, 400 static configurations of equilibrated systems werecollected. Then, the chemical potential of water beads μW ineach of 400 configurations was calculated using the test particleinsertion method of Widom.83 A test water bead was inserted ata randomly chosen location, 1 million trial insertions per frame.As a reference, the chemical potential of pure water μ0 wasdetermined by inserting a trial water bead into a cubic box of10Rc in size filled with water beads at reduced density ρRc

3 = 3.The saturation conditions of 100% humidity correspond to theequality of water chemical potentials in the membrane and inthe bulk water. The water humidity H that corresponds to agiven hydration level λ is determined by the difference of thesechemical potentials. Since one bead represents 4.5 molecules ofwater, the water activity is calculated as ln a(λ) = (μW(λ) −μW0)/(4.5kT). Assuming water vapor as an ideal gas, humidityis equal to the thus calculated activity.These calculations allowed us to construct the sorption

isotherms of water in Nafion, as given by our coarse-grainedmodels. The isotherms are presented in Figure 9. Theisotherms have type I shape by IUPAC classification character-istic of microporous adsorbents with a plateau at high humidity,which extends into the oversaturation region a > 1. Watercontent at a = 1 corresponds to sorption from saturated watervapor. Figure 9 shows that the higher the Meq, the higher theactivity at the same λ, and the lower hydration levelcorresponds to saturation, which is natural and agreesqualitatively with experimental observations.We noted that the beginning of saturation corresponds to the

formation of well-defined spheroidal water clusters discussed insection IV. Their growth coincides with a sharp increase ofactivity with λ. Within this regime, the chemical potential ofwater exceeds that in pure coarse-grained water (a > 1). Afterthat, the situation changes completely and activity becomesnearly independent of hydration level. We assume that thiscorresponds to the beginning of phase separation onto waterand hydrated Nafion, signified by growth of one or two largeclusters. These systems are typically thermodynamicallyunstable and a corresponding part of the isotherms havenegative slope dλ/da < 0, which can be noted in Figure 9.The adsorption isotherms from Figure 9 explain the non-

monotonic behavior of relative diffusivity found in NPT

simulations: decline in diffusion occurs at a thermodynamicallyunstable, unphysical region of oversaturation at a > 1, which isnot observed in experiment. In a homogeneous polymersolution, an oversaturated condition for the solvent (that ispossible as a metastable state with bulk separation prevented bya nucleation barrier) would not cause a decline of solventdiffusivity. Nafion, however, is segregated, and the phaseseparation into hydrated polymer and pure water, whenhydration exceeds the sorption capacity, is prevented byartificial periodic boundary conditions. The water clusternetwork connectivity depends strongly on the number of sidechains that concentrate at the interface between the hydrophilicand hydrophobic subphases. The growing domains of waterdrag the side chains to their surface, thus reducing the overallsurface area of the connecting channels and leading to theisolation of water clusters. This effect is artificial; it should notbe observed in a macroscopic system, as it was caused bylimited simulation time and system volume. As such, thesimulated dependencies of the water diffusivity on thehydration level should be considered only up to the saturationlevel (Figure 9).Despite the artificial nature of the structures with large

spherical water clusters, the reason why they do not merge isworth considering. As water bridges between the clusters “dryup” due to excessive surface area of the interface between thepolymer and water, large isolated clusters are formed. Eachcluster carries a positive charge due to the hydratedcounterions, which is compensated by the negative charge ofsulfonate groups surrounding the positively cluster. Thus, twoclusters experience electrostatic repulsion from each other,which creates a potential barrier and prevents their fusion.Because periodic boundary conditions suppress systemrestructuring and the simulation time is limited, the barriersassociated with the cluster coalescence never get overcome inmost of the simulations. Figure S4 in the SupportingInformation shows that the clusters experience significantfluctuations in shape but do not merge within the simulationrun.

VII. CONCLUSIONWe suggested a novel DPD model of hydrated Nafionmembranes. The proposed model accounts for polymer chainrigidity that is different for the Nafion backbone and side chainsand includes explicit electrostatic forces between the chargedpolymer fragments and dissociated counterions. In contrast toprevious works, we attempted to customize the coarse-grainedinteraction parameters against the available experimental andatomic simulation data. For the first time, we not only generateself-assembled structures in hydrated Nafion membranes withan experimentally informed DPD model and explore thespecifics of nanosegregated morphologies, but we also establishtheir thermodynamic properties, in particular, the relationshipbetween the level of hydration and humidity.The evolution of self-assembled morphology upon hydration

was studied with equilibrated NPT DPD simulations performedat a series of hydration levels λ from 2.25 to 18 water moleculesper sulfonate group. A classical percolation transition from asystem of isolated water clusters to a 3D network of hydrophilicchannels was observed. The hydrophilic subphase connectivitywas characterized by constructing its digitized replica andperforming random walk simulations to determine the effectivewater diffusivity. We found a significant decrease of waterdiffusivity in the membrane compared to the bulk water, e.g., by



2 orders of magnitude for Meq = 1144. These values arecomparable with the available experimental data. Noteworthy,at high hydration levels (e.g., above λ = 9 for the membrane ofequivalent molecular weight of 1144), we detected thecoalescence of water clusters into large domains, which werepoorly connected. This effect caused the reduction of the waterdiffusivity, which showed a maximum as a function of the levelof hydration. Using coarse-grained MC simulations, wecalculated the chemical potential of water in the hydratedpolymer and constructed the water sorption isotherms. Wedetermined that the maximum diffusivity and the onset offormation of large water domains corresponded to thesaturation conditions at 100% humidity and concluded thatthe oversaturated membrane morphologies generated in thecanonical ensemble DPD simulations corresponded to themetastable and unstable states that are not observed inexperiments. The hydration level of λ ≈ 7 at 100% humidityestablished by the proposed simulation method for Meq = 1144was comparable with available experimental data.This work also reveals serious limitations of the traditional

DPD models with an equal size of different types of beads andequal intracomponent repulsion parameters. First, this modeldoes not take into account different compressibilities of waterand polymer, and thus cannot predict correctly the elasticity ofthe polymer matrix and the membrane swelling. Note that themembrane swelling can be taken into account in the NPTsimulations with varying system volume. Second, the modeldoes not distinguish between different alkali cations consideredas “charged water beads”. Cations differ not only by their sizebut also by the size of the solvation shell that water formsaround them. This difference can be accounted for bycustomizing the cation bead diameter, charge smearing radiusRe, as well as the interactions with other beads, water beads inparticular. These parameters should be fitted to thermodynamicand kinetic properties of aqueous solutions of electrolytes, butsuch efforts have not been reported in the literature.It is desirable to extend the proposed DPD model of the

metal-substituted Nafion membrane to the acid formmembranes that are used in fuel cell and other electrochemicaldevices. However, this extension should take into account theeffects of deprotonation of sulfonate groups and protontransport within the hydrophilic subphase, including thehopping mechanism of proton diffusion. These problems willbe addressed in our future work.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information includes the following sections:(I) details of the calculation of bead volumes; (II) details ofMD simulations for parameter fitting; (III) details on DPDsimulations involved in parameter fitting, including specifics ofelectrostatic potentials and DPD simulations of hydratedNafion; (IV) tables with resulting water sorption, wateractivities, and related diffusivities and alternative versions ofselected figures in the logarithmic scale. This material isavailable free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected].

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported in parts by DTRA (grants HDTRA1-08-1-0042 and HDTRA1-14-1-0015) and NSF (grantDMR1207239).

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